Weak versus strong wave turbulence in the Majda-McLaughlin-Tabak model

We consider the one-dimensional Majda-McLaughlin-Tabak (MMT) model that describes the interactions of nonlinear and dispersive waves. We perform a detailed numerical study of the direct energy cascade. Our numerical experiments show the following. (i) In the limit of small nonlinearity the spectral slope observed in the statistical steady regime corresponds to the one predicted by the weak-wave-turbulence (WWT) theory. (ii) As the nonlinearity is increased, the WWT theory breaks down and deviations from its predictions are observed. (iii) It is shown that such departures from the WWT theoretical predictions are accompanied by the phenomenon of intermittency, typical of three-dimensional fluid turbulence. Our results clarify the role played by thewave-turbulence theory in the statistical description of nonlinear-dispersive waves described by the MMT model.